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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E40">Th42</font></span>: <a NAME="T42"><span class="comment"><font color="firebrick">:: MESFUN16:42</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b2">J</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/>  for <font color="Olive" title="b4">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a>,<a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>,<a href="dualsp02.html#K3" title="DUALSP02:func.3">RNS_Real</a><br/>  for <font color="Olive" title="b5">g</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><br/>  for <font color="Olive" title="b6">Pg2</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b3">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">J</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b2">J</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <font color="Olive" title="b4">f</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">I</font>,<font color="Olive" title="b2">J</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> &amp; <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> &amp; <font color="Olive" title="b6">Pg2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfun12.html#K3" title="MESFUN12:func.3">ProjPMap2</a> (<span class="p1">(<span class="default"><a href="mesfunc5.html#K1" title="MESFUNC5:func.1">R_EAL</a> <font color="Olive" title="b5">g</font></span>)</span>,<font color="Olive" title="b3">y</font>) holds <br/>( <font color="Olive" title="b6">Pg2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">I</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp; <font color="Olive" title="b6">Pg2</font> <a href="integra5.html#R1" title="INTEGRA5:pred.1">is_integrable_on</a> <font color="Olive" title="b1">I</font> )</div></div>
